Glasnik matematički, Vol. 54 No. 2, 2019.
Original scientific paper
https://doi.org/10.3336/gm.54.2.04
Generators and integral points on certain quartic curves
Yasutsugu Fujita
orcid.org/0000-0001-7985-9667
; College of Industrial Technology, Nihon University, 2-11-1 Shin-ei, Narashino, Chiba 275-8576, Japan
Tadahisa Nara
; Faculty of Engineering, Tohoku-Gakuin University, 1-13-1 Chuo, Tagajo, Miyagi 985-8537, Japan
Abstract
In this paper, we study integral points and generators on quartic curves of the forms \(u^2 \pm v^4=m\) for a nonzero integer $m$. The main results assert that certain integral points on the curves can be extended to bases for the Mordell-Weil groups of the elliptic curves attached to the quartic curves in the cases where the Mordell-Weil ranks are at most two. As corollaries, we explicitly describe the integral points on the quartic curves in each case where the ranks are one and two.
Keywords
Elliptic curve; quartic curve; canonical height; integral points
Hrčak ID:
229601
URI
Publication date:
11.12.2019.
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